Game apparatuses and methods for use in teaching the addition and subtraction of positive and negative numbers

ABSTRACT

A game apparatus and a method for assisting in teaching the addition and subtraction of positive and, especially, negative numbers, are based on the Null Theory of Adding and Subtracting Positive and Negative Numbers. The apparatus comprises (a) a plurality of positive units, (b) a plurality of negative units, (c) a demarcated playing zone, and (d) a means for measuring the number of free positive units and free negative units within the demarcated playing zone. Within the demarcated playing zone, free positive units combine with free negative units zone to form null units, with each null unit comprising an equal number of positive units and an equal number of negative units. Each null unit preferably comprises one positive unit and one negative unit.

FIELD OF THE INVENTION

[0001] The present invention relates to educational game apparatuses and methods for use in teaching the addition and subtraction of positive and, especially, negative numbers. The game apparatuses and methods of the present invention enable students to see and understand a theory for adding and subtracting positive and negative numbers.

DESCRIPTION OF THE PRIOR ART

[0002] A description of the prior art is set forth in U.S. Pat. No. 1,294,126, U.S. Pat. No. 3,094,792, U.S. Pat. No. 3,229,388, U.S. Pat. No. 3,410,002, U.S. Pat. No. 3,414,986, U.S. Pat. No. 3,452,454, U.S. Pat. No. 3,935,649, U.S. Pat. No. 4,177,681, U.S. Pat. No. 5,474,455, U.S. Pat. No. 6,089,871, and U.S. Pat. No. 6,413,099, which patents are incorporated herein in their entireties by reference.

[0003] As evidenced by the above-cited patents, educational game apparatuses and methods exist for teaching mathematical concepts. However, teaching the addition and subtraction of positive and, especially, negative numbers usually entails students learning by rote the rules of adding and subtracting positive and negative numbers without ever understanding the rhyme or reason behind what they are doing. Other students, unfortunately, never learn the rules and, for them, mathematics becomes a dreaded black hole.

SUMMARY OF THE INVENTION

[0004] Accordingly, a technique is needed for teaching the addition and subtraction of positive and, especially, negative numbers that clearly explains a cogent theory behind rules.

[0005] The apparatuses and methods of the present invention for teaching the addition and subtraction of positive and negative numbers solve the above need. More specifically, the present invention is based on the Null Theory of Adding and Subtracting Positive and Negative Numbers (hereinafter referred to as the “Null Theory”). According to the Null Theory, the natural state of a given environment is the null state. In the null state, the environment is in perfect balance and appears to be devoid of any matter. However, the environment is, in fact, composed of a plurality of null units, with each null unit being, in turn, composed of a positive unit and a negative unit. The environment can be disturbed by introducing (i.e., adding) into it one or more additional positive units or one or more negative units. The environment can also be disturbed by removing (i.e., subtracting) from it one or more positive units or one or more negative units. In the latter case, if there are not enough free positive units available to be removed from the environment, a sufficient number of null units are split to obtain the desired number of positive units to be removed from the environment. When a null unit is split and the positive unit thereof is removed from the environment, a negative unit is left behind in the environment. (Hence, the foregoing explanation of the Null Theory clarifies and visually demonstrates the reason behind the rule that a negative times a positive is a negative.) Likewise, if there are not enough free negative units available to be removed from the environment, a sufficient number of null units are split to obtain the desired number of negative units to be removed from the environment. When a null unit is split and the negative unit thereof is removed from the environment, a positive unit is left behind in the environment. (Accordingly, the foregoing explanation of the Null Theory clarifies and visually demonstrates the reason behind the rule that a negative times a negative is a positive.)

[0006] Another aspect of the Null Theory is that only excess positive units or excess negative units remain in the free state within the environment. For example, if there are 5 free positive units in the environment and if 3 negative units are introduced into the environment (as is the case in the mathematical expression 5+(−3)), the 3 negative units will combine with 3 of the free positive units to form 3 null units, leaving only 2 free positive units in the environment.

[0007] With the Null Theory in mind, in one embodiment of the present invention, the game apparatus employed to teach the addition and subtraction of positive and negative numbers comprises (a) a plurality of positive units, (b) a plurality of negative units, and (c) a demarcated playing environment or zone. The positive units and the negative units are adapted to reversibly attach to or be associated with one another to form null units, with each null unit comprising at least one positive unit and at least one negative unit and the number of positive units and the number of negative units per null unit being equal. (As used in the specification and claims, the terms “attached to” and “associated with” both mean that the objects in question either can be physically reversibly held together or can be positioned in a manner such that the objects appear to have an affinity for or relationship with one another.) Preferably, each null unit comprises just one positive unit and just one negative unit.

[0008] The demarcated playing zone is typically an integral part of a playing surface.

[0009] Generally, the game apparatus further comprising a means for measuring the number of units selected from the group consisting positive units, negative units, and combinations thereof, with the measuring means desirably being located on the playing surface and, preferably, within the demarcated playing zone. The measuring means, which can be a scale for weighting the positive and/or negative units, is ideally an axis marked in substantially equal units from 0 to M and in substantially equal units from 0 to N, where M is a positive whole integer, N is a negative whole integer, and substantially each of the positive units is adapted to reversibly attach to or be associated with a unit from 0 to M on the axis on the playing surface; and substantially each of the negative units is adapted to reversibly attach to or be associated with a unit from 0 to N on the axis on the playing surface. While M can be virtually any positive integer, M is typically a whole positive integer from 5 to 50, more typically from about 10 to 25, and most typically from 10 to 20. Similarly, while N can be virtually any negative integer, N is commonly a whole negative number from −5 to −50, more commonly from about −10 to −25, and most commonly from −10 to −20. Usually, M equals the absolute value of N.

[0010] It is also preferred that the game apparatus further comprising a plurality of means for reversibly holding a plurality of the null units located within the demarcated playing zone.

[0011] The game apparatus preferably also comprises an additional means for reversibly holding at least one positive unit and an additional means for reversibly holding at least one negative unit. More preferably, the game apparatus further comprises an additional means for reversibly holding a plurality of positive units and an additional means for reversibly holding a plurality of negative units. Most preferably, the additional positive unit holding means and the additional negative unit holding means are located on the playing surface outside the demarcated playing zone.

[0012] The playing surface of the game apparatus can be a screen of an electrical unit (such as a computer screen, a television screen, etc), a surface of a game board, a surface of a blackboard or other writing surface, a Velcro® surface, a static electricity charged surface, or the surface of any other apparatus or device capable of displaying the demarcated playing zone, the null units, the positive units, the negative units, and the means for showing the number of free positive units and/or free negative units within the demarcated playing zone.

[0013] The game apparatus of the present invention is employed in conjunction with a method that utilizes the principles of the Null Theory. In particular, the method of the present invention comprises the following steps:

[0014] Step A:

[0015] Play begins in a null state where a demarcated playing zone comprises a plurality of null units, with each null unit comprising at least one positive unit reversibly attached to at least one negative unit, the number of positive and negative units per null unit being equal.

[0016] Step B:

[0017] Take the first mathematical expression in a problem involving X mathematical expressions, with X being a whole positive number, and perform the mathematical operation indicated by the first mathematical expression. In the problem, the first mathematical expression is selected from the group of positive numbers and negative numbers and each of the remaining X−1 mathematical expressions is selected from the group consisting of the addition of a positive number, the subtraction of a positive number, the addition of a negative number, the subtraction of a negative number, and combinations thereof. In performing the mathematical operation indicated by the first mathematical expression, (i) if the first mathematical expression is a positive number Y₁, then the mathematical operation of step (B) comprises moving Y₁ free positive units from outside the demarcated playing zone to within the demarcated playing zone and (ii) if the first mathematical expression is a negative number Z₁, then the mathematical operation of step (B) comprises moving the absolute value of Z₁ free negative units from outside the demarcated playing zone to within the demarcated playing zone. (Alternatively, the mathematical operation indicated by the first mathematical expression can be performed as follows: (i) if the first mathematical expression is a positive number Y₁, then the mathematical operation of step (B) can be accomplished by breaking apart Y₁ null units into Y₁ free positive units and into Y₁ free negative units and moving the Y₁ free negative units from within the demarcated playing zone to outside the demarcated playing zone and (ii) if the first mathematical expression is a negative number Z₁, then the mathematical operation of step (B) can be accomplished by breaking apart the absolute value of Z₁ null units into the absolute value of Z₁ free positive units and into the absolute value of Z₁ free negative units and moving the absolute value of Z₁ free positive units from within the demarcated playing zone to outside the demarcated playing zone.)

[0018] Step C:

[0019] Take the second mathematical expression in the problem and perform the mathematical operation indicated by the second mathematical expression as follows:

[0020] 1. If the second mathematical expression represents the addition of a positive number Y₂, then step (C) comprises moving Y₂ free positive units from outside the demarcated playing zone to within the demarcated playing zone and, if there are any free negative units within the demarcated playing zone, then step (C) further comprises combining up to Y₂ free negative units with up to the Y₂ free positive units that were moved into the demarcated playing zone.

[0021] 2. If the second mathematical expression represents the subtraction of a positive number Y₂, then step (C) comprises moving Y₂ free positive units from within the demarcated playing zone to outside the demarcated playing zone and, if there are not Y₂ free positive units within the demarcated play zone to move to outside the demarcated zone, then step (C) further comprises breaking apart enough null units to obtain up to the required Y₂ free positive units and moving the Y₂ free positive to outside the demarcated playing zone.

[0022] 3. If the second mathematical expression represents the addition of a negative number Z₂, then step (C) comprises moving the absolute value of Z₂ free negative units from outside the demarcated playing zone to within the demarcated playing zone and, if there are any free positive units within the demarcated playing zone, then step (C) further comprises combining up to the absolute value of Z₂ free positive units with up to the absolute value of Z₂ free negative units that were moved into the demarcated playing zone.

[0023] 4. If the second mathematical expression represents the subtraction of a negative number Z₂, then step (C) comprises moving the absolute value of Z₂ free negative units from within the demarcated playing zone to outside the demarcated playing zone and, if there are not enough absolute value of Z₂ free negative units within the demarcated play zone to move to outside the demarcated playing zone, then step (C) further comprises breaking apart enough null units to obtain up to the required absolute value of Z₂ free negative units and moving the absolute value of Z₂ free negative to outside the demarcated playing zone.

[0024] Step D:

[0025] Repeat step (C) for each of the remaining X-2 mathematical expression in the problem.

[0026] As noted with respect to the game apparatus, in the method of the present invention, each null unit preferably comprises one positive unit reversibly attached to or associated with one negative unit.

[0027] It is also preferred that the method further comprise the step of measuring the number of the free positive units and the number of the free negative units within the demarcated playing zone. In one preferred version of the method of the present invention, the measurement is performed by the step of placing the free positive units that are within the demarcated playing zone along the positive portion of an axis marked with substantially equal spaces from 0 to M and the step of placing the free negative units that are within the demarcated playing zone along the negative portion of an axis marked with substantially equal spaces from 0 to N, where M and N are as previously defined above. While the positive portion of the axis preferably forms a continuum with the negative portion of the axis, the present invention includes the embodiment where there is a separate positive axis having substantially equal spaces from 0 to M and a separate negative axis having substantially equal spaces from 0 to N, where M and N are as previously defined.

[0028] While the foregoing apparatus and method can be used to teach young children who are just learning to add and subtract only positive numbers, when teaching the addition and subtraction of only positive numbers, it is sufficient to use a simpler game apparatus such as one comprising (a) at least one means for measuring unit increments; (b) a plurality of means for indicating a single unit; and (c) a means holding the plurality of single unit indicating means in slide able relationship to the measuring means. For instance, the plurality of single unit indicating means can comprise a plurality of beads, the holding means can comprise a dowel, with the beads being slide ably mounted on the dowel, and the game apparatus can comprises two measuring means (such as two rulers), with each measuring means being position so that the game apparatus can be played with equal facility by both right and left handed players.

[0029] For a fuller understanding of the nature and advantages of the mathematical game apparatuses and methods of the present invention, reference should be made to the ensuing detailed description taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Exemplary game apparatuses of the present invention are shown in the drawings where:

[0031]FIG. 1 is a top view of a playing surface employed in the method of the present invention;

[0032]FIG. 2 is an isometric view of a null unit employed in the method of the present invention showing a negative sign on one of the two constituent components of the null unit;

[0033]FIG. 3 is another isometric view of a null unit employed in the method of the present invention showing a positive sign on one of the two constituent components of the null unit;

[0034]FIG. 4 is an isomeric view of a null unit separated into its two constituent halves, namely, the positive component and the negative component;

[0035]FIG. 5 is a top view of a playing surface employed in the method of the present invention modified in accordance with the first mathematical expression of the mathematical problem of Example 1;

[0036]FIG. 6 is a top view of a playing surface employed in the method of the present invention modified in accordance with the second mathematical expression of the mathematical problem of Example 1;

[0037]FIG. 7 is a top view of a playing surface employed in the method of the present invention modified in accordance with the third mathematical expression of the mathematical problem of Example 1;

[0038]FIG. 8 is a top view of a playing surface employed in the method of the present invention modified in accordance with the fourth mathematical expression of the mathematical problem of Example 1;

[0039]FIG. 9 is a top view of a playing surface employed in the method of the present invention modified in accordance with the fifth mathematical expression of the mathematical problem of Example 1;

[0040]FIG. 10 is a perspective view of the front side of an apparatus within the cope of the present invention for teaching the addition and subtraction of just positive whole numbers; and

[0041]FIG. 11 is a perspective view of the backside of an apparatus within the scope of the present invention for teaching the addition and subtraction of just positive whole numbers.

[0042] It should be noted that the same numbers in the figures represent the same element of the game apparatuses of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0043]FIG. 1 shows an embodiment of the present invention in which the game apparatus 1 comprises a playing surface 2, a demarcated playing environment or zone 4 within the playing surface 2, a plurality of free positive units 6 removably attached to the playing surface 2 outside the demarcated playing zone 4, a plurality of free negative units 8 removably attached to the playing surface 2 outside the demarcated playing zone 4, a plurality of null units 10 removably attached to the playing surface 2 within the demarcated playing zone, and an axis 12 numbered from −10 to 10.

[0044] The playing surface 2 can be a game board, a computer screen, a television screen, a liquid crystal display screen, or any other means for displaying the demarcated playing zone 4, the free positive units 6, the free negative units 8, the null units 10, and the axis 12.

[0045] The area of the demarcated playing zone 4 can comprise a portion of the area of the playing surface 2 as shown in FIG. 1 or can comprise the entire area of the playing surface 2. When the area of demarcated playing zone 4 constitutes the entire area of the playing surface 2, the free positive units 6 and the free negative units 8 either can be held in essentially any container (such as a cup, a bowl, a bag, a pocket, a groove, etc.) or can be loose.

[0046] In the version of the invention shown in FIG. 1, the free positive units 6 and the free negative units 8 are removably attached to the playing surface 2 by snuggly fitting into depressions 14 in the playing surface 2. Other means for removably attaching the positive units 6 and the negative units 8 to the playing surface 2 include, but are not limited to, Velcro, snap fittings (such as used on clothing or on Lego® building blocks), screw fittings, gravity, etc. Of course, when the playing surface 2 is a computer screen or other essentially two-dimensional means for displaying the playing surface 2, the positive units 6 and negative units 8 are merely displayed on or merely displayed as being associated with the essentially two-dimensional means for displaying the playing surface 2.

[0047] Similarly, in the version of the invention shown in FIG. 1, the null units 10 are also removably attached to the demarcated playing zone 4 by snuggly fitting into depressions 16 in the demarcated playing zone 4. Likewise, other means for removably attaching the null units 10 to the demarcated playing zone 4 also include, but are not limited to, Velcro, snap fittings (such as used on clothing or on Lego® building blocks), screw fittings, gravity, etc. In addition, when the playing surface 2 is a computer screen or other essentially two-dimensional means for displaying the playing surface 2, the null units 10 are merely displayed on or merely displayed as being associated with the essentially two-dimensional means for displaying the playing surface 2.

[0048] In the embodiment of the invention shown in FIG. 1, the amount of free positive units 6 and free negative units 8 in the demarcated playing zone 4 are measured along the axis 12. As shown in FIG. 1, the axis 12 is numbered from 10 to −10. However, the axis could just as well be numbered from M to N, where M is a positive integer and N is a negative integer. Preferably, M is a positive integer from 5 to 50, more preferably from 10 to 25, and most preferably from 10 to 20. Similarly, N is a negative integer from −5 to −50, more preferably from −10 to −25, and most preferably from −10 to −20. Usually, M is equal to the absolute value of N. Furthermore, other means such as a weighting scale, a unit counter, etc. can be used to measure the number of free positive units 6 and free negative units 8 within the demarcated playing zone 4. Also, the measuring means can be located inside or outside the demarcated playing zone 4 and, in fact, on or off the playing surface 2. When the playing surface 2 is the display media used in conjunction with a computerized version of the method of the present invention, all that need be shown on the display media is the number of free positive units 6 and/or the number of free negative units 8 within the demarcated playing zone 4.

[0049] As shown in more detail in FIGS. 2 and 3, the null units 10 consist of a positive unit 6 removably attached to a negative unit 8. While the null unit can consist of a plurality of positive units 6 and a plurality of negative units 8 removably attached together, preferably the number of positive units 6 and the number of negative units 8 per null unit 10 are equal and, more preferably, the null unit consists of a single positive unit 6 removably attached to a single negative unit 8.

[0050] In the embodiment of the invention shown in FIG. 4, the positive unit 6 has a neck 18 that is adapted to snuggly fit into an opening 20 in the negative unit 8 to form the null unit 10. Other means for removably attaching the positive units 6 to the negative units 8 to form the null units 10 include, but are not limited to, Velcro, snap fittings (such as used on clothing or on Lego® building blocks), screw fittings, etc. In addition, when the playing surface 2 is a computer screen or other essentially two-dimensional means for displaying the playing surface 2, the positive unit 6 and negative unit 8 portions of each null unit 10 are merely displayed in close proximity (i.e., as being associated with one another) on the essentially two-dimensional means for displaying the playing surface 2.

[0051] The game apparatus of the present invention is employed in conjunction with a method that comprises the following steps:

[0052] Step A:

[0053] Play begins in a null state where a demarcated playing zone 4 comprises a plurality of null units 10, with each null unit 10 comprising at least one positive unit 6 reversibly attached to or associated with at least one negative unit 6, the number of positive units 6 and negative units 8 per null unit 10 being equal.

[0054] Step B:

[0055] Take the first mathematical expression in a problem involving X mathematical expressions, with X being a whole positive number, and perform the mathematical operation indicated by the first mathematical expression. In the problem, the first mathematical expression is selected from the group of positive numbers and negative numbers and each of the remaining X−1 mathematical expressions is selected from the group consisting of the addition of a positive number, the subtraction of a positive number, the addition of a negative number, the subtraction of a negative number, and combinations thereof. In performing the mathematical operation indicated by the first mathematical expression, (i) if the first mathematical expression is a positive number Y₁, then the mathematical operation of step (B) comprises moving Y₁ free positive units 6 from outside the demarcated playing zone 4 to within the demarcated playing zone 4 and (ii) if the first mathematical expression is a negative number Z₁, then the mathematical operation of step (B) comprises moving the absolute value of Z₁ free negative units 8 from outside the demarcated playing zone 4 to within the demarcated playing zone 4. (Alternatively, the mathematical operation indicated by the first mathematical expression can be performed as follows: (i) if the first mathematical expression is a positive number Y₁, then the mathematical operation of step (B) can be accomplished by breaking apart Y₁ null units 10 into Y₁ free positive units 6 and into Y₁ free negative units 8 and moving the Y₁ free negative units 8 from within the demarcated playing zone 4 to outside the demarcated playing zone 4 and (ii) if the first mathematical expression is a negative number Z₁, then the mathematical operation of step (B) can be accomplished by breaking apart the absolute value of Z₁ null units 10 into the absolute value of Z₁ free positive units 6 and into the absolute value of Z₁ free negative units 8 and moving the absolute value of Z₁ free positive units 6 from within the demarcated playing zone 4 to outside the demarcated playing zone 4.)

[0056] Step C:

[0057] Take the second mathematical expression in the problem and perform the mathematical operation indicated by the second mathematical expression as follows

[0058] 1. If the second mathematical expression represents the addition of a positive number Y₂, then step (C) comprises moving Y₂ free positive units 6 from outside the demarcated playing zone 4 to within the demarcated playing zone 4 and, if there are any free negative units 8 within the demarcated playing zone 4, then step (C) further comprises combining up to Y₂ free negative units 8 with up to the Y₂ free positive units 6 that were moved into the demarcated playing zone.

[0059] 2. If the second mathematical expression represents the subtraction of a positive number Y₂, then step (C) comprises moving Y₂ free positive units 6 from within the demarcated playing zone to outside the demarcated playing zone 4 and, if there are not Y₂ free positive units 6 within the demarcated play zone 4 to move to outside the demarcated zone 4, then step (C) further comprises breaking apart enough null units 10 to obtain up to the required Y₂ free positive units 6 and moving the Y₂ free positive units 6 to outside the demarcated playing zone 4.

[0060] 3. If the second mathematical expression represents the addition of a negative number Z₂, then step (C) comprises moving the absolute value of Z₂ free negative units 8 from outside the demarcated playing zone 4 to within the demarcated playing zone 4 and, if there are any free positive units 6 within the demarcated playing zone 4, then step (C) further comprises combining up to the absolute value of Z₂ free positive units 6 with up to the absolute value of Z₂ free negative units 8 that were moved into the demarcated playing zone 4.

[0061] 4. If the second mathematical expression represents the subtraction of a negative number Z₂, then step (C) comprises moving the absolute value of Z₂ free negative units 8 from within the demarcated playing zone 4 to outside the demarcated playing zone 4 and, if there are not enough absolute value of Z₂ free negative units 8 within the demarcated playing zone 4 to move to outside the demarcated playing zone 4, then step (C) further comprises breaking apart enough null units 10 to obtain up to the required absolute value of Z₂ free negative units 8 and moving the absolute value of Z₂ free negative units to outside the demarcated playing zone 4.

[0062] Step D:

[0063] Repeat step (C) for each of the remaining X−2 mathematical expression in the problem.

[0064] The method of the present invention is further illustrated in the following Example 1, which is intended to demonstrate, and not limit, the present invention:

EXAMPLE 1 Solve the Equation for x: 10+(−20)−(−15)−5+3=x

[0065] Initially, the playing surface 2 is in the null state as shown in FIG. 1. In the null state, there are no free positive units 6 or free negative units 8 in the demarcated playing zone 4.

[0066] The first step in solving the above equation is to write the equation in full as follows:

+10+(−20)−(−15)−(+5)+(+3)=x

[0067] In discussing the solution to foregoing equation, each element of the equation shall be identified as follows:

A S ₁ B S ₂ C S ₃ D S ₄ E

+10+(−20)−(−15)−(+5)+(+3)=x

[0068] With respect to element A, element A is the initial disturbed state of the demarcated playing zone 4. As shown in FIG. 5, the initial disturbed state of the demarcated playing zone 4 is achieved by moving 10 free positive units 6 a into the demarcated playing zone 4 from the playing surface 2 outside the demarcated playing zone 4. The 10 free positive units 6 a moved into the demarcated playing zone 4 are measured by placing them in the first 10 positive spaces along the axis 12.

[0069] Regarding the first mathematical operator S₁, S₁ is “+”, i.e., a positive sign, which denotes that element B is to be transported into (i.e., added to) the demarcated playing zone 4. Element B is −20 (i.e., 20 negative units 8). As shown in FIG. 6, 10 of the 20 negative units 8 that are moved into the demarcated playing zone combine with the 10 free positive units 6 a (see FIG. 5) already in the demarcated playing zone 4 to form 10 null units 10 b. The remaining 10 free negative units 8 b moved into the demarcated playing zone 4 are measured by placing them in the first 10 negative spaces along the axis 12.

[0070] As to the second mathematical operator S₂, S₂ is “−”, i.e., a negative sign, which denotes that element C is to be transported out of (i.e., subtracted from) the demarcated playing zone 4. Element C is −15 (i.e., 15 negative units 8). As shown in FIG. 7, 10 of the 15 negative units 8 c that are to be moved out of the demarcated playing zone 4 are present in demarcated playing zone 4 as free negative units 8 b of FIG. 6. The remaining 5 negative units 8 cc to be transported out of the demarcated playing zone 4 are obtained by taking 5 null units 10 from within the demarcated playing zone 4, breaking these 5 null units 10 into their component parts, namely, 5 free positive units 6 cc and 5 free negative units 8 cc, moving the thus obtained 5 free negative units 8 cc from the demarcated playing zone 4, and placing the remaining 5 free positive units 6 cc in the first 5 positive spaces along the axis 12.

[0071] With respect to the third mathematical operator S₃, S₃ is “−” i.e., a negative sign, which denotes that element D is to be transported out of (i.e., subtracted from) the demarcated playing zone 4. Element D is +5 (i.e., 5 positive units 6). As shown in FIG. 8, all of the 5 positive units 6 d that were moved out of the demarcated playing zone 4 were already present in the demarcated playing zone 4 as free positive units 6 cc shown in FIG. 7.

[0072] Regarding the fourth mathematical operator S₄, S₄ is “+”, i.e., a positive sign, which denotes that element E is to be transported into (i.e., added to) the demarcated playing zone 4. Element E is +3 (i.e., 3 positive units 6). As shown in FIG. 9, since there are no free negative units 8 in the demarcated playing zone 4, the 3 free positive units 6 e moved into the demarcated playing zone 4 are measured by placing them in the first 3 positive spaces along the axis 12.

[0073] Accordingly, x equals 3.

[0074] While the preferred embodiments of the invention have been set forth above in detail, some modifications can be made to the preferred version without departing from the spirit of the present invention. For example, while the above described game apparatus and method can be employed to teach students to add and subtract just positive numbers, a simpler device, such as the one shown in FIGS. 10 and 11 can be also be used. As shown in FIG. 10, the apparatus 30 comprises a body 32 having an opening 34 therein. In the opening 34 is mounted a dowel 36 and on the dowel are axially moveable beads 38. The number of axially movable beads 38 is generally from 5 to 25 and preferably from 10 to 20. On at least one side of the opening is a scale 40 (such as a ruler or other means for measuring length) for measuring the number of beads 38. The beads 38 can be moved to the left when they are being added and can be moved to the right when they are being subtracted.

[0075] The reverse side 42 of the apparatus 30 is shown in FIG. 11. In this view of this embodiment of the invention, all the elements mentioned in the preceding paragraph are also present. Accordingly, the apparatus 30 shown in of FIGS. 10 and 11 can be used with equal facility by left- and right-handed students.

[0076] Accordingly, the foregoing alternative embodiments are included within the scope of the present invention.

[0077] The apparatuses of the present invention can be made by techniques well know to those skilled in the art (e.g., injection molding, forged or cast metal, carpentry, etc.). In addition, software programs can be written by those skilled in the art for executing the methods of the present invention and the computerized versions of the invention can be played on a monitor of any suitably programmable electrical apparatus (such as a television screen, computer screen, liquid crystal display, etc.). 

What is claimed is:
 1. A game apparatus for assisting in teaching the addition and subtraction of positive and negative numbers, the game apparatus comprising: (a) a plurality of positive units; (b) a plurality of negative units; and (c) a playing surface, where the positive units and the negative units are adapted to reversibly be associated with one another to form a null means comprising at least one positive unit and at least one negative unit.
 2. The game apparatus of claim 1 where the playing surface comprises a demarcated playing zone.
 3. The game apparatus of claim 1 where the playing surface comprises a demarcated playing zone and the area of the playing surface and the area of the demarcated playing zone are substantially equal.
 4. The game apparatus of claim 1 further comprising a means for measuring the number of units selected from the group consisting positive units, negative units, and combinations thereof.
 5. The game apparatus of claim 1 further comprising a means for measuring the number of units selected from the group consisting positive units, negative units, and combinations thereof, where the playing surface comprises a demarcated playing zone and the measuring means is located within the demarcated playing zone.
 6. The game apparatus of claim 1 further comprising an axis, where the playing surface comprises a demarcated playing zone and the axis is located within the demarcated playing zone and marked in substantially equal units from N to M, where M is a positive whole integer; and N is a negative whole integer substantially each of the positive units is adapted to reversibly attach to a unit from 0 to M on the axis on the playing surface; and substantially each of the negative units is adapted to reversibly attach to a unit from 0 to N on the axis on the playing surface.
 7. The game apparatus of claim 1 further comprising a plurality of means for reversibly holding a plurality of the null means, where the playing surface further comprises a demarcated playing zone and the plurality of means for reversibly holding the plurality of the null means is located within the demarcated playing zone.
 8. The game apparatus of claim 1 where the null means comprises one positive unit and one negative unit reversibly attached together.
 9. The game apparatus of claim 1 further comprising an additional means for reversibly holding at least one positive unit and an additional means for reversibly holding at least one negative unit.
 10. The game apparatus of claim 1 further comprising an additional means for reversibly holding a plurality of positive units and an additional means for reversibly holding a plurality of negative units, where the playing surface comprises a demarcated playing zone and the additional positive unit holding means and the additional negative unit holding means are located on the playing surface outside the demarcated playing zone.
 11. The game apparatus of claim 1 where the playing surface comprises a demarcated playing zone and the game apparatus further comprises: (d) an axis located within the demarcated playing zone and marked in substantially equal units from 0 to M and in substantially equal units from 0 to N; (e) a plurality of means for reversibly holding a plurality of the null means located within the demarcated playing zone; and (f) an additional means for reversibly holding a plurality of positive units and an additional means for reversibly holding a plurality of negative units; where M is a positive whole integer; N is a negative whole integer substantially each of the positive units is adapted to reversibly attach to a unit from 0 to M on the axis on the playing surface; substantially each of the negative units is adapted to reversibly attach to a unit from 0 to N on the axis on the playing surface; the null means comprises one positive unit and one negative unit reversibly attached together; and the additional positive unit holding means and the additional negative unit holding means are located on the playing surface outside the demarcated playing zone.
 12. The game apparatus of claim 1 where the playing surface comprises a screen of an electrical monitor.
 13. The game apparatus of claim 1 where the playing surface comprises a surface of a game board.
 14. A method for adding and subtracting positive and negative numbers, the method comprising the steps of: (a) starting play in a null state where a demarcated playing zone comprises a plurality of null units, with each null unit comprising at least one positive unit reversibly associated with at least one negative unit, the number of positive and negative units per null unit being equal; (b) taking the first mathematical expression in a problem involving X mathematical expressions where the first mathematical expression is selected from the group of positive numbers and negative numbers and each of the remaining X−1 mathematical expressions are selected from the group consisting of the addition of a positive number, the subtraction of a positive number, the addition of a negative number, the subtraction of a negative number, and combinations thereof, where X is a positive whole integer and performing the mathematical operation indicated by the mathematical expression whereby (i) if the first mathematical expression is a positive number Y₁, then step (b) comprises moving Y₁ free positive units from outside the demarcated playing zone to within the demarcated playing zone and (ii) if the mathematical expression is a negative number Z₁, then step (b) comprises moving the absolute value of Z₁ free negative units from outside the demarcated playing zone to within the demarcated playing zone; (c) taking the second mathematical expression in the problem and performing the mathematical operation indicated by the second mathematical expression whereby (i) if the second mathematical expression represents the addition of a positive number Y₂, then step (c)(i) comprises moving Y₂ free positive units from outside the demarcated playing zone to within the demarcated playing zone and, if there are any free negative units within the demarcated playing zone, then step (c)(i) further comprises combining up to Y₂ free negative units with up to the Y₂ free positive units that were moved into the demarcated playing zone; (ii) if the second mathematical expression represents the subtraction of a positive number Y₂, then step (c)(ii) comprises moving Y₂ free positive units from within the demarcated playing zone to outside the demarcated playing zone and, if there are not Y₂ free positive units within the demarcated play zone to move to outside the demarcated zone, then step (c)(ii) further comprises breaking apart enough null units to obtain up to the required Y₂ free positive units and moving the Y₂ free positive to outside the demarcated playing zone; (iii) if the second mathematical expression represents the addition of a negative number Z₂, then step (c)(iii) comprises moving the absolute value of Z₂ free negative units from outside the demarcated playing zone to within the demarcated playing zone and, if there are any free positive units within the demarcated playing zone, then step (c)(iii) further comprises combining up to the absolute value of Z₂ free positive units with up to the absolute value of Z₂ free negative units that were moved into the demarcated playing zone; (iv) if the second mathematical expression represents the subtraction of a negative number Z₂, then step (c)(iv) comprises moving the absolute value of Z₂ free negative units from within the demarcated playing zone to outside the demarcated playing zone and, if there are not the absolute value of Z₂ free negative units within the demarcated play zone to move to outside the demarcated playing zone, then step (c)(iv) further comprises breaking apart enough null units to obtain up to the required absolute value of Z₂ free negative units and moving the absolute value of Z₂ free negative to outside the demarcated playing zone; and (d) repeating step (c) for each of the remaining X−2 mathematical expression in the problem.
 15. The method of claim 14 where step (a) comprises starting play in the null state where the demarcated playing zone comprises a plurality of null units, with each null unit comprising one positive unit reversibly associated with one negative unit.
 16. The method of claim 14 further comprising the step of measuring the number of the free positive units and number of the free negative units within the demarcated playing zone.
 17. The method of claim 13 further comprising the step of placing the free positive units that are within the demarcated playing zone along the positive portion of an axis marked with substantially equal spaces from 0 to M and the step of placing the free negative units that are within the demarcated playing zone along the negative portion of the axis marked with substantially equal spaces from 0 to N, with M being a positive whole integer and N being a negative whole integer.
 18. The method of claim 13 where step (a) comprises starting play in the null state where the demarcated playing zone comprises a plurality of null units, with each null unit comprising one positive unit reversibly associated with one negative unit and where the method further comprises the step of placing the free positive units that are within the demarcated playing zone along the positive portion of an axis marked with substantially equal spaces from 0 to M and the step of placing the free negative units that are within the demarcated playing zone along the negative portion of the axis marked with substantially equal spaces from 0 to N, with M being a positive whole integer and N being a negative whole integer.
 19. A method for adding and subtracting positive and negative numbers, the method comprising the steps of: (a) starting play in a null state where a demarcated playing zone comprises a plurality of null units, with each null unit comprising at least one positive unit reversibly associated with at least one negative unit, the number of positive and negative units per null unit being equal; (b) taking the first mathematical expression in a problem involving X mathematical expressions where the first mathematical expression is selected from the group of positive numbers and negative numbers and each of the remaining X−1 mathematical expressions are selected from the group consisting of the addition of a positive number, the subtraction of a positive number, the addition of a negative number, the subtraction of a negative number, and combinations thereof, where X is a positive whole integer and performing the mathematical operation indicated by the mathematical expression whereby (i) if the first mathematical expression is a positive number Y_(x), then the mathematical operation of step (b) comprises breaking apart Y₁ null units into Y₁ free positive units and into Y_(x) free negative units and moving the Y₁ free negative units from within the demarcated playing zone to outside the demarcated playing zone; and (ii) if the first mathematical expression is a negative number Z₁, then the mathematical operation of step (b) comprises breaking apart the absolute value of Z₁ null units into the absolute value of Z₁ free positive units and into the absolute value of Z₁ free negative units and moving the absolute value of Z₁ free positive units from within the demarcated playing zone to outside the demarcated playing zone; (c) taking the second mathematical expression in the problem and performing the mathematical operation indicated by the second mathematical expression whereby (i) if the second mathematical expression represents the addition of a positive number Y₂, then step (c)(i) comprises moving Y₂ free positive units from outside the demarcated playing zone to within the demarcated playing zone and, if there are any free negative units within the demarcated playing zone, then step (c)(i) further comprises combining up to Y₂ free negative units with up to the Y₂ free positive units that were moved into the demarcated playing zone; (ii) if the second mathematical expression represents the subtraction of a positive number Y₂, then step (c)(ii) comprises moving Y₂ free positive units from within the demarcated playing zone to outside the demarcated playing zone and, if there are not Y₂ free positive units within the demarcated play zone to move to outside the demarcated zone, then step (c)(ii) further comprises breaking apart enough null units to obtain up to the required Y₂ free positive units and moving the Y₂ free positive to outside the demarcated playing zone; (iii) if the second mathematical expression represents the addition of a negative number Z₂, then step (c)(iii) comprises moving the absolute value of Z₂ free negative units from outside the demarcated playing zone to within the demarcated playing zone and, if there are any free positive units within the demarcated playing zone, then step (c)(iii) further comprises combining up to the absolute value of Z₂ free positive units with up to the absolute value of Z₂ free negative units that were moved into the demarcated playing zone; (iv) if the second mathematical expression represents the subtraction of a negative number Z₂, then step (c)(iv) comprises moving the absolute value of Z₂ free negative units from within the demarcated playing zone to outside the demarcated playing zone and, if there are not the absolute value of Z₂ free negative units within the demarcated play zone to move to outside the demarcated zone, then step (c)(iv) further comprises breaking apart enough null units to obtain up to the required absolute value of Z₂ free negative units and moving the absolute value of Z₂ free negative to outside the demarcated playing zone; and (d) repeating step (c) for each of the remaining X−2 mathematical expression in the problem.
 20. A game apparatus for assisting in teaching the addition and subtraction of positive numbers, the game apparatus comprising: (a) at least one means for measuring unit increments; (b) a plurality of means for indicating a single unit; and (c) a means for holding the plurality of single unit indicating means in slide able relationship to the measuring means.
 21. The game apparatus of claim 20 where the plurality of single unit indicating means comprise a plurality of beads, the holding means comprises a dowel, the beads are slide ably mounted on the dowel, and the game apparatus comprises two measuring means, with each measuring means being position so that the game apparatus can be played with equal facility by both right and left handed players.
 21. The game apparatus of claim 19 where the plurality of single unit indicating means comprise a plurality of beads, the holding means comprises a dowel, the beads are slide ably mounted on the dowel, and the game apparatus comprises two measuring means, with each measuring means comprising a ruler and each ruler being position so that the game apparatus can be played with equal facility by both right and left handed players. 